Relative weak compactness and three sequential notions

Let $X$ be a real or complex Banach space, give it its weak topology $\sigma (X,X^{\ast })$ from Weak topology on a normed space, and let $A\subseteq X$.

• $A$ is relatively weakly compact if its weak closure ${\bar{A}}^w$ is weakly compact. It is weakly compact if $A$ itself, with the relative weak topology, is compact.
• $A$ is relatively weakly sequentially compact if every sequence $(a_n{)}_{n\in \mathbb{N}}$ in $A$ has strictly increasing indices $n_0<n_1<\cdots$ and a point $x\in X$ such that $a_{n_k}\to x$ weakly. It is weakly sequentially compact if the limit can always be taken in $A$.
• $A$ is relatively weakly countably compact if every sequence $(a_n)$ in $A$ has a weak cluster point $x\in X$, meaning that for every weak neighborhood $U$ of $x$ and every $N\in \mathbb{N}$ there is an $n\ge N$ with $a_n\in U$. It is weakly countably compact if the cluster point can always be taken in $A$.

The cluster-point condition is indexed: a value occurring infinitely often is a cluster point even when the range of the sequence is finite. Thus constant and eventually constant sequences have the expected cluster point. The empty set satisfies all three relative conditions: its weak closure is empty and compact, and there is no sequence with values in it. In fact the corresponding absolute conditions are vacuous or compact for the same reason.

These are definitions, not implications between the notions. In a general topological space the three properties need not coincide. Their equivalence for weak subsets of Banach spaces is the content of Eberlein–Šmulian later on this page. “Relative” permits a sequential limit or cluster point in the ambient $X\setminus A$; “absolute” does not.